it doesn't matter that the confirm page does not have a form
the action of the form on the contact_us.php page is set to post to the contact-us-confirm.php page:
<form action="contact-us-confirm.php" method="post" name="form2" id="form2">
This means that when the form is submitted, it immediately passes the form information to the contact-us-confirm.php page. all processing of the form data must occur on that page.
The validation of the contact_us form must occur on the contact-us-confirm.php page since that is where the action tells the server where to send the form data.